Download A model of interacting partons for hadronic structure functions

A Model of Interacting Partons
for Hadronic Structure Functions 1
arXiv:hep-ph/9911538 v1 30 Nov 1999
by
Govind S. Krishnaswami
Advisor: Professor S.G.Rajeev
Department of Physics and Astronomy, University of Rochester
Rochester, NY 14627, USA
Submitted as a senior thesis to the University of Rochester [in partial fulfillment of the requirements for the degree Bachelor of Science in Physics], May
1999
1
Received the American Physical Society’s Apker Award for 1999.
Abstract
We present a model for the structure of baryons in which the valence partons interact
through a linear potential. This model can be derived from QCD in the approximation
where transverse momenta are ignored. We compare the valence quark distribution function
predicted by our model with that extracted from global fits to Deep Inelastic Scattering
data. The only parameter we can adjust is the fraction of baryon momentum carried by
valence partons. Our predictions agree well with data except for small values of the Bjorken
scaling variable.
1
Contents
1 Introduction
3
2 Deep Inelastic Scattering
7
2.1
The Quark Model and Quantum ChromoDynamics . . . . . . . . . . . . . .
8
2.2
Deep Inelastic Scattering and the Parton Model . . . . . . . . . . . . . . . .
10
2.3
Null Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
3 The Interacting Valence Parton Model
14
3.1
The Quark-Quark Potential . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
3.2
Wave Function of the Proton . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
3.3
Hamiltonian of the N-Parton System . . . . . . . . . . . . . . . . . . . . . .
16
3.4
The Hartree Ansatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
3.5
Momentum Sum Rule and Boundary Conditions . . . . . . . . . . . . . . . .
19
3.6
Positivity of the Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
3.7
The Energy in Momentum Space . . . . . . . . . . . . . . . . . . . . . . . .
21
3.8
Integral Equation for the Ground State Wave Function . . . . . . . . . . . .
23
3.9
Parameters in the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
4 Determination of the Valence Quark Wave Function
25
4.1
Estimation of the Ground State using Variational Ansatzes . . . . . . . . . .
25
4.2
Non-relativistic Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
4.3
Finite Part Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
4.4
The Small p Behavior of the Valence Quark Wave Function . . . . . . . . . .
30
4.5
Numerical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
4.6
Comparison with Experimental Data . . . . . . . . . . . . . . . . . . . . . .
33
5 Conclusion
35
2
1
Introduction
This thesis is about the structure of the proton. The proton, along with the neutron is one
of the constituents of the atomic nucleus. For a long time since their discovery, it was not
known whether the proton and neutron had substructure, and if so, what their constituents
were like. The Deep Inelastic Scattering Experiments [1] of the early 1970s found that the
proton was made of point-like constituents called quarks or partons.
These experiments were similar in spirit to the alpha particle scattering experiments
of Rutherford, which established that the atom contains a point-like nucleus. He scattered
alpha particles against a thin gold foil and found that there was a small probability for them
to scatter through wide angles. This would not be possible if the positive charge in an atom
was uniformly distributed. Moreover, the scattering cross-section he measured was that of
a point-like nucleus carrying all the positive charge of the atom! Subsequent experiments
showed that the nucleus was made of protons and neutrons. Charge radii measured in
elastic electron-proton scattering showed that the proton was not elementary. What were
its constituents like?
The Deep Inelastic Scattering experiments scattered electrons against protons. This
time, the scattering was inelastic. The inclusive scattering cross-section was measured and
expressed in terms of ‘structure functions’. The structure functions describe the structure
of the proton. These experiments used the electromagnetic force between the electron and
the constituents of the proton to study the ‘strong force’ that held the proton together. The
electromagnetic force is mediated by the exchange of a photon. By making the wave-length
of the photon small enough, it was possible to ‘look’ deep within the proton.
The startling discovery of the Deep Inelastic Scattering Experiments was that as long
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as one looked closely enough, the structure functions did not depend on the wave-length of
the photon! The constituents of the proton did not have any length-scale associated with
them! This phenomenon is called scaling: the constituents of the proton appeared point-like.
The parton model was proposed by Bjorken, Feynman and others [2] as a simple
explanation of scaling in Deep Inelastic Scattering. The proton was thought of as being
made of point-like constituents called partons. These were identified with the quarks which
were until then, hypothetical particles. Structure functions can be expressed in terms of
parton distribution functions. These describe the distribution of partons within the proton.
Soon after, Quantum ChromoDynamics (QCD), the fundamental theory of strong interactions was discovered. Scaling was understood as a consequence of asymptotic freedom in
QCD [3]. Small violations of scaling were predicted by perturbative QCD. These have been
confirmed by experiments. However, the initial parton distributions cannot be calculated
within perturbative QCD. They have as yet not been understood theoretically. However,
these parton distributions are so important to high energy hadron collisions that they have
been measured and extracted from experimental data in great detail [4, 5, 6].
Understanding the structure functions of quarks in a proton is not unlike understanding
the orbits of planets around the sun or the wavefunctions of electrons in an atom. Understanding the latter has proven extremely fruitful in all of science. In the case of the proton,
we are dealing with the strong force rather than the gravitational or electromagnetic.
In this thesis, we shall present a model of interacting partons to explain the distribution of valence quarks in the proton. The quarks are assumed to be relativistic particles
interacting with each other through a linear potential in the ‘null’ coordinate. Their momenta transverse to the direction of the collision will be ignored in favour of the much larger
longitudinal momentum. That collinear QCD can describe hadronic structure functions has
also been proposed by others [7]. The ground state wave function of the proton will be
the one that minimizes its energy. We will approximate the proton wave function with a
product of single parton wave functions. This is analogous to the Hartree approximation in
Atomic Physics. These approximations will allow us to determine the parton wave functions
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as the solution to a non-linear integral equation. We shall study this equation in several
ways (both analytic and numerical) and obtain a fairly complete quantitative picture of the
valence quark distribution.
Finally we will compare our predictions with the parton distributions extracted from
experimental data. Considering that we have at-most one parameter to adjust (the fraction
f, of proton momentum carried by the valence quarks) the agreement is quite good, except
for small values of the momentum fraction
xB . In this region, our model is not expected
to be accurate. We may not ignore sea-quarks, gluons and transverse momenta.
Wave Functions Versus xB
3
<---MRST fit to DATA
2.5
2
<---Numerical f~.6
1.5
<---Analytic f~.5
1
0.5
0.2
0.4
0.6
0.8
Figure 1. Comparison of the predicted valence parton wavefunction
1
√
xB
φ(xB ) with the MRST [5]
global fit to data. The wavefunction we predict goes to a non-zero constant at the origin. The
‘analytic’ prediction is obtained as a variational approximation. The numerical solution is not
reliable in this region of small xB . The fit to data has a mild divergence at xB = 0.
These results have been described in a recent publication [8]. Furthermore, this interacting
parton model can be derived from QCD [9] in the limit where transverse momenta are ignored.
The question of deriving the spectrum and structure functions of hadrons from QCD is an
old and important one. The meson spectrum and wave functions of two dimensional QCD were
obtained by ’t Hooft [10] in 1974 by a clever summation of planar Feynman diagrams in the large N
limit. Perturbation theory works in the case of mesons since they are described by small fluctuations
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from the vaccuum. However, the baryons remained elusive. Later, Witten [11] suggested that the
baryon can be described by a Hartree Fock approximation in the large N limit of QCD. He carried
out this idea in a non-relativistic context. In the early 1980s, Skyrme’s [12] idea that the baryon
is a topological soliton was revived by Balachandran and others [13] and shown to be consistent
with QCD. Rajeev [9] developed Quantum HadronDynamics (QHD) in two dimensions. Two
dimensional QHD is an equivalent reformulation of two dimensional QCD in terms of observable
particles: the hadrons, rather than the quarks, which are confined within the hadrons. Baryons
are the topological solitons of QHD while mesons are small fluctuations of the vaccuum. The large
N limit of 2dQHD reproduces [9] the meson spectrum and wave functions of ’t Hooft. But it also
allows us to predict the structure of the baryon. Within a variational approximation, the structure
of the baryon can be estimated by a non-linear integral equation [9]. It was found that this integral
equation also had a simple derivation in terms of the parton model [8, 9]. We present this parton
model point of view here.
This thesis is organised into 3 main chapters. Chapter 2 provides background, a discussion
of Deep Inelastic Scattering and the parton model. In Chapter 3 we present the interacting parton
model and derive the integral equation satisfied by the valence quark wave function. Chapter 4
discusses how we solve for the valence quark wave function and comparison with experimental
data. We do not assume much familiarity with particle physics. We will often use terms that are
specialized to this branch of physics, but most of them are defined or explained at some point.
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2
Deep Inelastic Scattering
In electron-proton Deep Inelastic Scattering for instance, a high energy electron scatters against a
proton. The electron does not feel the strong force and its weak interaction is dominated by the
electromagnetic interaction, which is mediated by the exchange of a high energy virtual photon.
Thus the kinematics is described in terms of two 4-vectors: q µ the momentum of the photon and
pµ the momentum of the proton. The photon momentum is space-like, q 2 < 0 while pµ is timelike, p2 = Mp2 > 0. Mp is the rest mass of the proton. The scattering is inelastic. The proton
disintegrates producing several hadrons in the final state:
eP → eX
If we sum over all possible final states X, we get the inclusive deep inelastic cross-section, which
is expressed in terms of two dimensionless scalar ‘structure functions’ F1 and F2 . Being Lorentz
scalars, they can depend only on p2 , p.q and q 2 . p2 = Mp2 is fixed by the mass of the proton. For
convenience, we may take our two independent scalars as Q2 = −q 2 , and the dimensionless ratio
xB =
Q2
2p.q ,
called the Bjorken scaling variable.
So F1,2 = F1,2 (xB , Q2 ).
Q2 , being the square of the momentum transferred by the photon, sets the energy scale of
the experiment. Q is inversely proportional to the wave length of the photon. An experiment at
large Q2 is therfore looking deep inside the proton. If Q2 >>
1
a
1
a2 ,
we are in the deep inelastic region.
∼ 100M eV is the inverse charge radius of the proton in its rest frame.
In the deep inelastic region, we will show that xB can be thought of as the fractional mo-
mentum of the quark inside the proton that scatters the photon. This will be made clear in the
context of the parton model to be discussed soon.
The Deep Inelastic Scattering experiments of the early 1970s [1] showed that at sufficiently
large Q2 >> Λ2QCD , the structure functions were approximately independent of Q2 . They depended,
7
not on the Lorentz Scalars Q2 or 2p.q separately, but only on their ratio xB . This phenomenon is
called Bjorken scaling.
The structure functions may be expressed in terms of parton distribution functions [14].
They directly describe the structure of the proton. The parton distribution functions φa (xB , Q2 ) are
momentum space probability distribution functions. For instance, the up quark parton distribution
function gives the probability density of finding an up quark carrying a fraction xB of proton
momentum, when the proton is probed at the energy scale Q2 . As with the structure functions, the
parton distribution functions are essentially independent of Q2 . Being probability distributions,
they are the absolute squares of the parton wave functions inside the proton. The reduction from
structure functions to parton distribution functions is made possible by ignoring certain correlations
and the factorization theorem of perturbative QCD [15, 14]. Roughly speaking, the amplitude for
the virtual photon to scatter off the proton is expressed as a sum of products of two factors: the
amplitude for it to scatter elastically off a quark of given momentum and the probability of finding
a quark with that momentum in the proton. While the elastic scattering off a quark of a given
momentum is calculable in perturbation theory, the probability of finding such a quark in the
proton is non-perturbative and yet to be calculated from first principles.
2.1
The Quark Model and Quantum ChromoDynamics
The quark model, based on hadron spectroscopy suggested that for the purpose of quantum numbers, the proton is a colorless combination of 2 flavours of quarks: up, up and down. The quarks
are fermions. In addition to quantum numbers like position, spin and charge, quarks carry flavour
(up, down, strange etc.) and color quantum numbers. Color is the analogue of electric charge in
the theory of strong interactions. The number of colors N is 3 in nature. Quantum ChromoDynamics (QCD) is the presently accepted theory of the color force. The strong force is mediated
by the exchange of spin 1 bosons called gluons. It was developed partly in analogy with Quantum
ElectroDynamics (QED) which is the theory of the electromagnetic force. QCD is a quantum field
theory with a non-abelian gauge symmetry. While its basic framework is understood, it has proven
very hard to solve and make predictions about non-perturbative effects.
The proton is just one of several particles which take part in strong interactions. They are
collectively called hadrons. Hadrons are broadly classified according to their spin. Integer spin
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bosonic hadrons are called mesons while the half integer spin fermions are called baryons. The
proton and neutron are the lightest and most common baryons. The pions are the lightest mesons.
Free quarks have never been observed in nature: they are confined within the hadrons, which are
colorless combinations of quarks.
In the late 1960s and early 70s, the parton model for hadrons was proposed by Bjorken,
Feynman and others [2] to explain the phenomenon of scaling in Deep Inelastic Scattering. According to the original parton model, in the deep inelastic region, the proton behaved as though it
was made up of essentially free point-like constituents called partons. These partons were identified
with the quarks (u, u and d in the case of the proton).
When probed at even higher energies (larger Q2 ) and small xB , it was found that there
is a small probability of finding anti-quarks (such as u and d ) and even quarks of other flavours
(strange quarks for instance) inside the proton. These did not fit directly into the original parton
model. However, these additional probability distributions were measured and are described in
terms of a phenomenological extension of the original parton model. We now speak of valence, sea,
anti-quark and gluon distributions in the proton. The valence quarks are the 3 quarks (up, up and
down in the case of the proton) of the original parton model. They are named in a loose analogy
with the valence electrons of the atom. One needs to probe deeper into the proton (higher Q2 )
before ‘sea’ quarks and anti-quarks become significant.
In this paper we shall primarily be interested in developing a model that predicts the
xB
dependence of valence quark wave functions. Since this is the main goal, we shall ignore some other
details and small corrections, which though important and measurable, obscure the main point.
For instance, the wave function depends on the isospin of the baryon (proton or neutron). We
shall calculate a single valence quark wave function ignoring isospin effects, which must therefore
be compared to the isoscalar combination of experimentally measured up and down quark wave
functions in the baryon. Correcting for isospin effects is not hard, but will not be addressed here.
In the physics of strong interactions, our experimental knowledge and precision of measurements far exceeds our theoretical understanding in most areas. In contrast with QED, where theory
and experiment agree to an unprecedented degree, in the non-perturbative regime of QCD we are
barely at the qualitative or 10% level!
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2.2
Deep Inelastic Scattering and the Parton Model
Let us now return to the kinematics of deep inelastic collisions. Up to now we described the
scattering in Lorentz invariant language. To understand the parton model of Feynman better, it
will be useful to consider the ‘infinite-momentum’ frame.
Figure 2. A parton model schematic of a Deep Inelastic Collision. Figure taken from [16].
In this inertial reference frame, the proton has a very large 3-momentum P and we may
ignore its rest mass in comparison pµ = (|P|, P). The proton is thought of as consisting of several
point-like constituents, the partons. The 4-momentum of a given parton is expressed as xpµ where
0 ≤ x ≤ 1 is the fraction of proton momentum carried by the given parton. Deep Inelastic
Scattering is visualized as initially, the elastic scattering of the photon against a parton. The
partons are assumed to have negligible mass m. This is followed by a complicated process of
hadronization in which the partons recombine to form several hadrons in the final state X.
From our point of view, that of predicting parton distribution functions, there is an important
simplification that Deep Inelastic Collisions allow. The component of momentum in the direction
of the collision far exceeds those transverse to it. Therefore, it is reasonable to ignore the transverse
components of the parton momenta. It is this approximation that makes the problem of predicting
hadronic structure functions tractable from a theoretical point of view. We are essentially dealing
with a problem in 2 space-time dimensions.
The parton model also gives us a very useful physical interpretation of the Bjorken scaling
variable, xB . xB was one of the two Lorentz scalars used to parametrize the structure functions.
We will show that xB =
Q2
2p.q
is the same as the fraction x of proton momentum p carried by the
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parton that scatters off the photon. Momentum conservation implies that (xp + q)2 = m2 where
p is the 4-momentum of the proton, q the momentum of the photon and m the mass of the struck
quark. But m is negligibly small and x2 p2 = x2 Mp2 where Mp is the mass of the proton. In the
deep inelastic region, Q2 >> Mp2 and therefore x ∼
Q2
2p.q
which we recognize as the Bjorken scaling
variable xB . Therefore we may interpret xB in the deep inelastic region as the fraction of proton
4-momentum carried by a parton.
This parton model interpretation of xB as the fraction of proton momentum carried by a
parton suggests that 0 ≤ xB ≤ 1 at least in the deep inelastic region. This is in fact true in
general. It follows from conservation of 4-momentum in a collision and the fact that the proton
is the lightest baryon. Since xB is a Lorentz scalar, it may be evaluated in any inertial reference
frame. In the lab frame, a photon of 4-momentum q µ = (ν, q) collides inelastically with a proton
of mass Mp at rest. The invariant (mass)2 of the final state X is W2 .
Figure 3. Kinematics of a Deep Inelastic Collision in the lab frame. Figure taken from [16].
Even in an inelastic collision, 4-momentum is conserved:
[(ν, q) + (M, 0)]2 = W 2
Therefore,
q 2 + 2M ν + M 2 = W 2 , where
q 2 = q µ qµ . Writing
Q2 for
−q 2 we see that
Q2 = 2M ν − (W 2 − M 2 ) . Now the proton is the lightest baryon, while the final state X includes
several heavier hadrons, hence
transfer ν ≥ 0 and
W 2 ≥ M 2 and
Q2 ≥ 0, we have
xB =
Q2
2M ν
≤ 1 . Moreover, since the energy
xB ≥ 0. Therefore, 0 ≤ xB ≤ 1 as desired.
While we are still in the lab frame, it will be good to point out precisely what the deep
inelastic limit is. We have xB =
Q2
2M ν .
The deep inelastic region of the xB − Q2 parameter space is
the limit of large Q2 and ν keeping xB fixed.
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In the deep inelastic limit, the structure functions are essentially independent of Q2 . They
depend only on xB . For instance, F2 falls by about 50% as Q2 increases from 1 to 25 GeV 2 ,
while the square of the elastic form factor falls by a factor of 106 over the same range! [16]. This
invariance of the Structure Functions on the energy scale of the measurement, Q2 is known as
Bjorken Scaling and was first observed in the Deep Inelastic Scattering Experiments of the 1970s
[1]. The proton when probed at high energies (small distances), appeared to consist of point-like
constituents.
Figure 4. F2 as a function of Q2 at xB = .25. For this choice of xB , there is practically no
Q2 dependence of the structure function, that is, exact scaling. (After Friedman and Kendall [1]
(1972).) Figure taken from [16].
Soon, however, it was found that scale invariance was only an approximate symmetry: the
structure functions had a very slow (logarithmic) Q2 dependence. The accurate prediction of the Q2
dependence of the structure functions is one of the major successes of perturbative QCD. However,
there is as yet no satisfactory theoretical understanding of the xB dependence of the structure
functions. It is essentially controlled by non-perturbative effects. The fundamental theory of
strong interactions QCD has been in place for several years now, but has proven very hard to solve
even approximately. We shall describe a model which allows us to calculate the xB dependence
of hadronic structure functions. This model can be derived as an approximation to the large N
limit of two dimensional QCD. N here is the number of colors. It was actually obtained as an
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approximation to Quantum HadronDynamics(QHD), an equivalent reformulation of 2dQCD in
terms of color invariant observables [9]. However, we are able to interpret and describe this model
in terms of the parton model. It is this parton model point of view that we shall describe in what
follows.
Before describing the interacting parton model, let us introduce a convenient coordinate
system in two space-time dimensions: the null coordinates.
2.3
Null Coordinates
It will prove extremely convenient to use a null coordinate system when describing the kinematics in
the valence parton model. One useful consequence of this is that the relativistic energy-momentum
p
dispersion relation E = ± p21 + m2 will no longer involve complications due to the square-
roots. The sign of energy will be the same as that of momentum! In QHD, the field variable
M(p,q) depends on two space-time points that are separeated by a null distance and thus causally
connected.
As explained earlier, we will ignore the transverse momenta of the partons. We may take
the longitudinal momenta of the partons to be directed along the x1 axis. The two dimensional
space-time position and momentum in cartesian coordinates are (x0 , x1 ) and (p0 , p1 ). Rather than
use p1 as the kinematic variable, we will use the null component of momentum p = p0 − p1 . ‘p’
is the momentum along a null line in 2-dimensional Minkowski space. We will use (p0 , p) as our
coordinates. This is not an orthogonal coordinate sytem. However, the simplification achieved is
that the mass-shell condition for a particle of mass m: p20 = p21 + m2 is replaced by p0 = 12 (p +
m2
p )
where p = p0 − p1 is the null momentum.
Thus we see that for a particle with positive energy p0 , the null momentum must be positive,
unlike in cartesian coordinates. Also note that the fraction x of proton 4-momentum carried by a
parton is the same as the fraction of corresponding null momenta. For brevity, we shall often refer
to null momentum just as momentum. There should be no confusion since we will no longer use
the cartesian coordinate p1 .
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3
3.1
The Interacting Valence Parton Model
The Quark-Quark Potential
We now describe a model for the structure of baryons in the language of the parton model. In the
original quark-parton model, the baryon is made of N partons, which are essestially massless, free
particles. N here is the number of colors, which is 3. We shall keep N arbitrary for now. It is
true that when probed at very small distances (large momentum transfers, Q2 ), the quarks inside
the proton appear to be free particles. This is called asymptotic freedom [3] in QCD. However,
quarks are confined within the proton. An isolated quark has never been produced. Therefore, even
though the force that binds quarks together is vanishingly small at small distances, it is non-zero
at intermediate distances and does not diminish at large distances. This is in stark contrast to the
electromagnetic and gravitational forces which decrease with distance!
If the partons in our model were indeed free, they would not bind to form a proton. The
only way the proton can have non-trivial structure functions is for the quarks to interact. Since we
are describing a many-body system (N partons), it is simplest to assume an attractive two body
potential between the quarks.
In Quantum Chromodynamics, the force between quarks is mediated by the exchange of
massless spin 1 bosons called gluons. Gluons are the carriers of the color force just as photons are
the quanta of the electromagnetic force. When QCD is dimensionally reduced to 1+1 dimensions
[9], the gauge fields may be eliminated using their equations of motion. What remains is effectively
a long range quark-quark force which is given by a linear two-body potential in the null coordinates.
This linear potential can also be understood in other ways. Since gluons are massless, their
propogator in momentum space is
1
q2 .
This is the same as the photon propogator. In 4 space-time
dimensions, the photon propogator corresponds to the
1
|r−r ′ |
Coulomb potential. However, since
we are in 2 space-time dimensions, we must use the appropriate Coulomb potential, which is the
14
Green’s function of the Laplace operator (second derivative with respect to x) in 1 space dimension.
This is 21 |x − y|. Notice that this potential is linear, attractive and confining.
Moreover, there is evidence that the quark-quark potential is linear as reported in [17]. This
interacting parton model can be derived as an approximation to Quantum HadronDynamics (QHD)
[8, 9]. In QHD, Lorentz invariance leads to the choice of a linear potential. Translation invariance
implies that it can only depend on the absolute value of the separation
3.2
|x − y| .
Wave Function of the Proton
We assume that the proton is made up of N partons (quarks). Therefore, the proton is a relativistic
quantum many-body system. We will describe the proton in terms of a wave function that will
depend on the quantum numbers of the partons (momenta, spin, color and flavour.)
We are interested in knowing the baryon wave function in its ground state, since that is where
the proton spends most of its time. The strategy will be to write down the energy of the baryon
in the state ψ and minimize this energy with respect to different choices of ψ. The configuration ψ
which minimizes the energy is its ground state wave function. This is what we will compare with
experimental data.
The quarks are spin
1
2
fermions transforming under the fundamental representation of the
color group SU (N ). However, the proton itself is colorless. (i.e. a color singlet) It must transform
under the trivial representation of SU (N ). As mentioned earlier, we will work in the null coordinates. Therefore the wave function of a single parton is ψ̃(a, α, p). Here ‘a’ labels the flavour
(up, down) and the spin quantum numbers of the parton. α labels color and p = p0 − p1 is the
null momentum of the parton. We use ψ̃ to denote the momentum space wave function. The
corresponding position space wave function ψ(a, α, x) is the inverse Fourier transform of ψ̃(a, α, p).
Since the null momentum is always positive, ψ̃ must vanish for negative p. We shall often refer to
this as ‘ψ̃ has only positive support’.
We can think of ψ̃ as a joint probability density. It is a probability density that depends
on two discrete random variables ‘a’ and ‘α’ and one continuous random variable p, for any given
parton. The proton is made up of N partons and has a wave function
ψ̃(a1 , α1 , p1 ; · · · ; aN , αN , pN )
Here ai , αi , pi are the spin, flavour, color and null momentum of the ith parton. Since the
15
quarks are fermions, the proton wave function should be completely anti-symmetric under interchange of a pair of quarks, by the Pauli exclusion principle. However, since the baryon is invariant
under color, the wave function must be completely anti-symmetric in color alone.
ψ̃(a1 , α1 , p1 ; · · · ; aN , αN , pN ) =
ǫα1 ,···,αN
√
N!
ψ̃(a1 , p1 ; · · · ; aN , pN )
This means the wave function is completely symmetric in spin, flavour and momentum quantum numbers. Therefore, once color has been factored out, the partons behave like a collection of
bosons.
3.3
Hamiltonian of the N-Parton System
What is the energy of the proton in the state ψ̃(a1 , p1 ; · · · ; aN , pN )? We will express the energy
as a sum of kinetic and potential energies. Consider the case N = 1,i.e. a proton consisting of
a single parton with wave function ψ̃(a, p). The ‘effective mass’ of the parton is µa , which could
possibly depend on what its flavour is. We shall return to a discussion of the effective mass later.
The relativistic kinetic energy-momentum relation expressed in terms of null-momentum is
p0 = 12 (p +
µ2a
p ).
The probability density that the parton indeed has null momentum p and spin-flavour ‘a’ is
|ψ̃(a, p)|2 . Therefore, the expectation value of the kinetic energy of this single parton baryon is
P R∞
a 0
1
2 (p
+
µ2a
2 dp
p )|ψ̃(a, p)| 2π
It follows that the kinetic energy of a baryon with N partons is
P
a1 ,a2 ,···,aN
R ∞ PN
0
1
i=1 2 (pi
+
µ2ai
2 dp1 ···dpN
pi )|ψ̃(a1 , p1 ; · · · ; aN , pN )|
(2π)N
(We shall choose our normalization constants for Fourier transforms so that a factor of
1
2π
consis-
tently appears in the measure of every momentum space integral.)
The potential energy is most easily expressed in position space. As explained earlier, the
partons interact through a two-body potential. i.e. the potential energy of a pair of quarks at null
coordinates x and y is g2 v(x − y), where g is a coupling constant with the dimensions of mass. The
potential
v is linear in the separation between the quarks v(x − y) =
potential energy
16
1
2 |x
− y|. This yields a
g̃ 2
2
P
a1 ,...,aN
R∞ P
−∞
i6=j
v(xi − xj )|ψ(a1 , x1 ; · · · ; aN , xN )|2 dx1 · · · dxN
for the system of N partons. The factor of
1
2
is to avoid double counting. ψ(a1 , x1 ; · · · ; aN , xN ) is
the position space wave function of the baryon. It is the inverse Fourier transform of the momentum
space wave function.
ψ(a1 , x1 ; · · · a1 , xN ) =
R∞
0
ψ̃(a1 , p1 ; · · · aN , pN )ei
P
j
pj xj dp1 ···dpN
(2π)N
We said that µa is the effective mass of ‘a’ parton of flavour a inside the proton. This must
be explained. A quark is confined within the proton and cannot be isolated as a bare particle.
Therefore, its bare mass is not a directly measurable quantity. All that matters is its effective
mass µa when confined within the proton. However, one can make an indirect measurement of
of the ‘bare’ quark masses. At high energies, the strong force is essentially zero and the electroweak interactions of these essentially free quarks depends on their bare masses. These have been
measured and are called ‘current’ quark masses. The current quark mass of the up and down
quarks, which are the valence quarks in the proton, are about 5 and 8 MeV/c2 . This must be
contrasted with the energy scale of the strong interactions ΛQCD ∼ 100 MeV. We see that the bare
up and down quark masses are negligible in comparison. In QCD, the limit of zero quark mass
is the limit of Chiral Symmetry. It is probably best to think of the quark mass parameters ma
as measuring the extent to which Chiral Symmetry is broken in the theory. In nature, the quark
masses are not exactly zero, but they are zero to a good first approximation. It is this limit that
we shall mostly be interested in while comparing our results with experimental data.
Returning to the question of effective and bare quark masses, we mention that a similar
situation exists in the case of an electron in a metal. It has an effective mass that is different
from the mass of a free electron. The term in the energy that involves the effective mass may
be re-expressed as the sum of a term involving the bare mass alone and a term involving the
coupling constant. This latter term is often referred to as the self-energy term. In QED, which is
an interacting theory of the electron, the ‘bare’ electron can emit and re-absorb photons. This part
of the interaction can be thought of as renormalizing the bare mass of the electron, giving rise to
a self-energy
However, these analogies are only to set the context. They cannot be pushed too far, in the
present case, the square of the effective mass is infact negative in the limit of chiral symmetry! In
17
this case, the self energy term really arises in 2 dimensional QCD when a quartic operator in the
potential energy is normal ordered [9]. The resulting expression for the effective mass in terms of
2
the bare mass is µ2a = m2a − g̃π , where g̃2 = N g2 . This expression has been derived in the literature
[10, 18].
Adding the potential and kinetic energies, the total energy of the proton is
EN (ψ̃) =
a1 ···aN
1
+ g2
2 a
N
∞X
µ2
dp1 · · · dpN
1
[pi + ai ]|ψ̃(a1 , p1 ; · · · aN , pN )|2
2
pi
(2π)N
0
i=1
Z
∞ X
X
v(xi − xj )|ψ(a1 , x1 ; · · · aN , xN )|2 dx1 · · · dxN .
X Z
1 ···aN
−∞ i6=j
The wave function that minimizes this energy subject to the constraint that the total probability is 1 is the ground state baryon wave function. Here the norm of ψ̃ is given by
||ψ̃||2 =
3.4
P
a1 ,···,aN
R∞
0
1 ···dpN
|ψ̃(a1 , p1 ; · · · ; aN , pN )|2 dp(2π)
N
The Hartree Ansatz
The minimization of the above energy can be thought of as a problem in many-body theory. We
don’t have a general solution to such a problem and must look for approximation methods. An
important simplification can be made since we are describing the ground state of a bosonic system,
since we have factored out the totally anti-symmetric color part of the wave function. Mean field
theory gives a good approximation to the ground state where each boson moves in the mean field
created by the others. Moreover, in the ground state, each of the bosons can be assumed to occupy
the same single particle state. Therefore, in the mean field approximation, we make the ansatz:
P
Q
ψ̃(a1 , p1 ; · · · ; aN , pN ) = δ( i pi − P ) N
i=1 ψ̃(ai , pi ).
In the language of probability distributions, we are assuming that the joint probability distribution of the N partons is well approximated by the product of N independent (identical) single
parton distributions. Since the null momenta are positive we impose the condition that the null momenta of the partons add up to that of the baryon (P). Therefore, the single parton wave functions
must vanish for p < 0 and p > P . (Infact, the condition is weaker, all we need is the delta function
18
factor on the momentum sum above. But the difference is a correlation which is suppressed in
the limit of a large number of partons, N.) Thus we are ignoring quark-quark correlations. When
comparing with data, we will express the wave function in terms of the dimensionless fractional
momentum x =
p
P.
In Section 2.2 we showed that this fractional momentum is nothing but the
Bjorken scaling variable xB in the deep inelastic limit.
In practice, this is a very reasonable assumption. It underlies most of the phenomenological
description of Deep Inelastic Scattering. The very fact that one can speak of an up quark momentum
distribution in the proton, without mentioning the simultaneous down quark momentum at which
it was measured means that it is a good approximation to ignore correlations.
Such a mean field approximation is common in many-body physics and is probably best
known in the context of the Hartree approximation of atomic physics. There, we are solving the
Schrödinger equation for a many electron atom. The number of electrons in a neutral atom is
the same as the atomic number Z. The mean field approximation can be thought of as a large-Z
approximation [9]. The fact that it works well even for Helium when Z = 2 encourages us to use it
here for N = 3. For a more precise analogy, see the appendix to [9].
3.5
Momentum Sum Rule and Boundary Conditions
There is one further constraint that the single particle wave function must satisfy: the momentum
sum rule. The expectation value of the total momentum of all the valence partons must equal the
fraction f of the baryon momentum actually available to them
N
RP
0
dp
= f P.
p|ψ̃(p)|2 2π
The valence quarks do not carry all the momentum of the baryon, as in the original parton model.
The parton distributions extracted from Deep Inelastic Scattering data show that roughly half the
momentum of the baryon is carried by the valence quarks. The rest is in the gluons, sea quarks
and anti-quarks. We shall see that the fraction
f is the only parameter on which the parton
distributions predicted by our model depend.
The energy per parton is therefore:
E =
XZ
a
0
P
µ2
dp
1
[p + a ]|ψ̃(a, p)|2 +
2
p
2π
19
1 2
g̃
2
Z
∞
−∞
v(x − y)
X
a
|ψ(a, x)|2
X
β
|ψ(β, y)|2 dxdy.
with the potential v(x − y) = 12 |x − y| as explained in Section 3.1. The single parton wave function
is determined by minimizing the energy per parton subject to the constraints: ||ψ̃|| = 1 and
ψ̃(a, p) = 0 unless 0 ≤ p ≤ P . We will impose the boundary condition that the wave function
vanish as p → P . This is because the probability density of a single parton carrying all momentum
of the baryon must be zero. The behavior of the wave function as p → 0 will be worked out in
detail in the next chapter (Section 4.4). It will turn out that for a positive quark mass m > 0, the
wave function vanishes at the origin too. In the limiting case m = 0, the wave function goes to a
non-zero value at the origin.
The condition that the momentum space wave function vanish outside [0,P] looks rather
different in position space. It says that ψ(a, x) must be the value along the real line, of an entire
function. This constraint can be hard to implement especially if we look for numerical solutions.
Hence we work in momentum space.
We also make a comment on the support of the wave function. We have imposed the constraint that ψ̃(p) vanish for p > P . i.e. the null momentum of a parton must not exceed that of
RP
dp
= f P , where f is the
the baryon. However, the momentum sum rule states that N 0 p|ψ̃(p)|2 2π
fraction of baryon momentum carried by the N valence partons. But this is just N times the mean
(first moment) of the single parton distribution! We see that the single parton wave function is
primarily concentrated around the region p =
fP
N .
For large enough N, it should be reasonable
to allow the wave functions to be non-zero beyond P, since they are vanishingly small for large p
anyway. Thus we may replace the upper limit of integration P, by ∞ especially while obtaining
analytic approximations. Note that we are not saying that P → ∞. This is not a Lorentz invariant statement. P is the null momentum of the baryon and has a fixed value in any given inertial
reference frame.
3.6
Positivity of the Energy
We may reformulate this constrained minimization problem as the solution of a certain non-linear
integral equation. Before proceeding with this, let us make some slight simplifications. We will set
all the parton masses equal to each other. In nature, the up and down current quark masses are
20
both roughly zero. Further, let us look for a single parton wave function that is non-zero for only
a single spin-flavour index ‘a’ : ψ̃(a, p) = δa,1 ψ̃(p).
This breaks the U(M) invariance of our model spontaneously. This symmetry can be restored
later by the collective variable method as in the theory of solitons, though we shall not address
these issues here.
With these simplifications, the energy is given by:
Z P
Z
1
µ2
1 2 ∞
2 dp
E =
[p + ]|ψ̃(p)|
+ g̃
v(x − y)|ψ(x)|2 |ψ(y)|2 dxdy.
2
p
2π
2
0
−∞
We take a moment to check that the energy is positive (and hence bounded below), so that its
minimization is a well posed problem! This is immediate since the integrands of both the kinetic
and potential energies are positive functions for µ2 = m2 −
g̃ 2
π
≥ 0. The physically interesting
region is the limit of zero quark mass m → 0. This corresponds to a negative value of µ2 . This
RP
R∞
2
dp
case is more subtle, the kinetic ( 0 21 [p + mp ]|ψ̃(p)|2 2π
) and potential energies ( 12 g̃2 −∞ v(x −
RP
2
2 dp
y)|ψ(x)|2 |ψ(y)|2 dxdy) remain positive while the self energy term ( 0 12 [ −g̃
πp ]|ψ̃(p)| 2π ) is negative.
However, it can be shown that this self energy term is cancelled by a portion of the potential energy,
when expressed in momentum space. In practice, this will not be a problem since we will approach
the physically interesting region from positive values of µ2 and see that a sensible limit exists.
3.7
The Energy in Momentum Space
The constraint on the position-space wave function is that it be the boundary value of an entire
function. The same constraint is much simpler in momentum space: it must vanish outside the
interval [0,P]. Therefore, as mentioned in Section 3.5, it will be convenient to express the potential
energy in momentum space. The energy per parton is
E =
Z
P
0
µ2
dp 1 2
1
[p + ]|ψ̃(p)|2
+ g̃
2
p
2π 2
Z
∞
−∞
v(x − y)|ψ(x)|2 |ψ(y)|2 dxdy.
The potential energy is best understood in the language of electrostatics. It is just the
electro-static potential energy of a charge density ρ(x) = g̃|ψ(x)|2 in one space dimension, where
the Green’s function is
1
2 |x
− y|. g̃ here is analogous to the unit of electric charge. The Poisson
integral formula then gives the electrostatic potential
21
V (x) = g
R∞
2 |x−y| dy
2
−∞ |ψ(y)|
Rather than refer to V(x) as the electrostatic potential, we must call it the mean potential
due to the partons. It is not to be confused with the 2-body potential v(x − y) between the
quarks! V(x) then satisfies Poisson’s equation V ′′ (x) = g̃|ψ(x)|2 . The boundary conditions are
R∞
g̃|x|
V (0) = g̃ −∞ |ψ(y)|2 |y|
2 dy and as |x| → ∞, V (x) → 2 , which is just the asymptotic form of the
potential of a charge g̃ localized around the origin.
We shall rewrite all of this in momentum space, but will have to be careful since we will
find that the mean potential Ṽ (p) has a
−1
p2
singularity at the origin in momentum space. The
momentum space integrals we get will therefore be singular and we will define them as Finite Part
integrals in the sense of Hadamard [19]. These ‘Finite Part’ prescriptions are not to be thought
of as a witch-craft that allows one to assign finite values to divergent integrals, but as a way of
restating the above boundary conditions in momentum space. Ultimately, it is the physics that
determines what the correct definition of an apparently divergent quantity is. As we shall see, these
definitions will turn out to be very natural and motivated by general principles such as analytic
continuation. It is therefore likely that these are the ‘correct’ definitions in other contexts too.
We first rewrite Poisson’s equation for the mean potential in momentum space by Fourier
RP
dq
. We see that
transforming: −p2 Ṽ (p) = W̃ (p) where W̃ (p) is the convolution 0 ψ̃(p + q)ψ̃(q) 2π
the mean potential is singular at the origin: Ṽ (p) →
−1
p2
as p → 0. It will be useful to note that
W̃ (0) = 1, Ṽ (−p) = Ṽ ∗ (p) and W̃ ′ (0) = − 21 |ψ̃(0)|2 . Also, by a choice of phase, the wave function
ψ̃(p) may be taken to be real. Therefore Ṽ (p) is real and even.
Therefore, the energy in momentum space is:
E[ψ̃] =
RP
0
1
2 (p
+
µ2
2 dp
p )|ψ̃(p)| 2π
where Ṽ (p) =
+ 2g̃ FP
−g̃
p2
RP
0
R
dpdq
Ṽ
(2π)2
(p)ψ̃(q)ψ̃ ∗ (q − p)
dr
ψ̃ ∗ (p + r)ψ̃(r) 2π
We see that the integrand in the potential energy has a
1
p2
singularity and we must define it as a
‘Finite Part’ (FP) integral. We will postpone a discussion of Finite Part integrals till we actually
have to evaluate them in the next chapter.
22
3.8
Integral Equation for the Ground State Wave Function
We must minimize the above energy functional with respect to the constraint that the norm of ψ
be 1. This is equivalent to minimizing F [ψ̃] = E[ψ̃] + λ||ψ̃||2 . λ here is the Lagrange multiplier
enforcing the constraint. Since the functional we are minimizing is quartic in ψ, λ is not the same as
the ground state energy. Therefore the groundstate wave function satisfies the equation
δF [ψ̃]
δψ̃
= 0,
which when written out more explicitly reads
1
2 (p
+
µ2
p )ψ̃(p) +
g̃F P
RP
0
dq
Ṽ (p − q)ψ̃(q) 2π
= λψ̃(p)
Since Ṽ is quadratic in ψ̃, we see that this is a non-linear singular integral equation for ψ̃.
In the next chapter, we will explain how the singular integrals are defined and obtain a
solution to the equation. We must mention that there is a well-developed theory of linear integral
equations, even those with Cauchy-type singularies [19]. But the non-linearities and severity of
the singularities in our case mean that we will have to develop special techniques to deal with our
integral equation.
The strategy will be to understand the ground state wave function from several different
points of view. No single technique may give us the exact solution. We will develop approximation
methods which are valid in distinct special cases. By patching these together, we will obtain both
a qualitative and quantitative understanding of the solution.
3.9
Parameters in the Model
Let us conclude this chapter with a discussion of the parameters in the theory. g̃ is the coupling
constant of the theory. It has dimensions of mass and is related to the coupling constant of twodimensional QCD by g̃ = g2 N . g is related to the coupling constant of the four dimensional theory
through a multiplicative factor of the order of the inverse transverse size of the proton. This arises
when we ‘integrate over’ transverse coordinates to arrive at this ‘effective’ two-dimensional model.
Though we do not predict the value of the coupling constant, we will see that it cancels out when
we calculate the valence quark distribution.
N is the number of colors, which is the same as the number of valence quarks. Most of the
approximations we have made become exact when N → ∞. We will use the value N = 3 when
comparing with data.
23
2
µ is the ‘effective’ quark mass and is related to the current quark mass by µ2 = m2 − gπ . We
will be interested in the limit of chiral symmetry when m → 0. The energy expressed above has
logarithmic divergences in both the potential and self energies when m is exactly zero. However, we
shall define the theory as the limit m → 0. A Frobenius analysis will show that the valence quark
wave function has a sensible limit as m → 0. This can be thought of as an infrared regulation. But
we emphasize that there is no real divergence in the problem, requiring renormalization. In any
event, nature is balanced on a knife-edge at a small, positive value of m!
The limit where m is small keeping g̃ fixed is the extreme relativistic limit. The binding
energy of the system far exceeds the rest-mass of the partons. We will also have occasion to
consider the non-relativistic limit in which µ >> g̃. This is not the physically interesting case as
it corresponds to a baryon made up of heavy quarks. However, in this limit the integral equation
will reduce to a non-linear ordinary differential equation and will give us another way of studying
the solutions.
λ is the Lagrange multiplier enforcing the constraint on the norm of ψ̃. It has no direct
physical meaning and is not equal to the energy of the state ψ̃.
P is the null momentum of the baryon, which in its restframe, is equal to its rest-mass. ψ̃ is
non zero only for 0 ≤ p ≤ P. But when we express the wave function in terms of the dimensionless
Lorentz invariant variable xB =
p
P,
we will see that the structure functions do not depend on which
value of P we used (i.e. which inertial reference frame we picked). g̃ too will cancel out. One
can think of P as being measured in units of g̃. When the dimensionless structure functions are
expressed in terms of the dimensionless variable xB , they no longer depend on g̃.
The only free parameter in the theory then, is the fraction f of baryon momentum carried
RP
dp
by the valence partons. It enters through the momentum sum rule N 0 p|ψ̃(p)|2 2π
= fP.
24
4
4.1
Determination of the Valence Quark Wave Function
Estimation of the Ground State using Variational Ansatzes
The variational principle of quantum mechanics allows us to estimate the ground state energy and
wave function of a quantum mechanical system. The ground state is the one that actually minimizes
the energy. Therefore, by minimizing the energy within a sub-class of functions in the Hilbert space,
we may obtain an upper bound for the ground state energy and also a qualitative understanding
of the ground state wave function. In practice, a judicious choice of a 1 or 2 parameter family of
variational ansatzes motivated by physical principles can often come spectacularly close to the true
ground state energy.
The energy per parton is given by
E =
Z
P
0
µ2
dp 1 2
1
[p + ]|ψ̃(p)|2
+ g̃
2
p
2π 2
Z
∞
−∞
v(x − y)|ψ(x)|2 |ψ(y)|2 dxdy.
As a first approximation, we may work on the interval [0, ∞) with a function that decays as
p → ∞, as justified in Section 3.5. A simple choice is ψ̃a (p) = Cpe−ap with a > 0, p ≥ 0. Here ‘a’
is a variational parameter controlling the rate of decay of the wave function in momentum space.
C is fixed by normalization. In position space, this positive momentum function has an analytic
continuation to the upper half plane and a double pole at x = −ia: ψ(x) =
′
C
.
(x+ia)2
We may calculate the energy of such a state and minimize with respect to a. It is clear on
R∞
dp
dimensional grounds that the term 0 12 p|ψ̃(p)|2 2π
scales like a1 while all other terms in the energy
scale like a. (‘a’ has dimensions of inverse momentum or length.) Thus there is a value of ‘a’ at
which the energy is minimized among this class of functions.
Eground
state
≤
1
2
p
I1 (m2 I2 + g̃2 I3 ),
25
where I1 , I2 and I3 are positive dimensionless pure numbers obtained from the integrations.
We have also tried another class of variational ansatzes: ψ̃(p) = Cpα(1 − p)β . This time, we
work on the finite interval [0, P ] and set P = 1 for convenience. Thus p is the fraction of baryon
momentum carried by the quark. This is the same as the Bjorken Scaling variable xB . The energy
calculations in this case are more formidable but can be performed in terms of hypergeometric
functions. Moreover, we work in momentum space and use the definitions of singular integrals,
which will be presented in Section 4.3. The results, however, are encouraging and easy to state.
Among the choices α, β = 1, 2, 3 we find that the energy is minimized for the choice α = 1 and
β = 2, even as m → 0. There is a further decrease in energy when we consider wave functions
of the form ψ̃(p) = Cpν (1 − p)2 for fractional values of α, 0 < α < 1. Our conclusion is that in
the limit of chiral symmetry (zero current quark mass), the energy is minimized in the limit where
α → 0. We thus have a qualitative understanding of the momentum space wave function as m → 0.
It goes to a non-zero constant as p → 0 and then falls off roughly as (1 − p)2 as p → 1.
We shall see that this picture is re-inforced and made more precise by the other approximation
methods we develop to study the integral equation for ψ̃.
4.2
Non-relativistic Limit
The non-relativistic limit is the limit where the binding energy of the system is small compared to
the rest mass of the quarks. It can be achieved by making the quark mass large compared to the
coupling constant (m >> g̃). In the case of the proton, this is not a very interesting limit since
the up and down quark masses are essentially zero. However, we will be able to understand our
system from a slightly different perspective this way. Besides, it also gave us a way of checking our
numerical procedures. The expression for the energy is:
E=
RP
1
0 2 (p
+
µ2
2
p )|ψ̃(p)|
+
g̃ 2
2
R∞
2 |x−y| |ψ(y)|2 dxdy.
2
−∞ |ψ(x)|
The relativistic energy-null momentum dispersion relation is p0 = 21 (p +
µ2
p ),
p > 0. We see
that p0 has a minimum when p = µ. In the extreme case of no interaction at all (g̃ = 0), the total
energy is just the kinetic energy. In this case the energy is minimized when all the null-momentum
of the parton is concentrated at the value p = µ. In other words, |ψ̃(p)|2 is a delta function at
p = µ.
26
The effect of having a small, but non-zero interaction g̃ > 0 is to broaden out the ground
state wave function. In any case, it is concentrated around the minimum of the dispersion curve.
Therefore we may replace the dispersion relation by its quadratic approximation in the neighbourhood of its minimum, p = µ. The result should not be surprising. We get the new dispersion
relation
p0 = µ + k2 /2µ,
where k = p − µ is the shifted momentum variable and µ =
heavy quark.
q
m2 −
g2
π
∼ m is the mass of the
We immediately recognize this as the non-relativistic energy of a particle of mass µ and
momentum k. Putting this into the expression for the energy per parton, we have
E =µ+
R
k2
2 dk
2µ |χ̃(k)| 2π
+
g̃
2
R∞
−∞ V
(x)|χ(x)|2 dx.
where χ̃(k) = ψ̃(µ + k) and χ(x) is its Fourier transform. V(x) is the effective potential
satisfying Poisson’s equation with boundary conditions as in Section 3.7:
V ′′ (x) = g|χ(x)|2
To minimize this energy we vary with respect to χ(x) and impose the constraint ||χ|| = 1.
The result is a Schrödinger-like equation.
−1 ′′
2µ χ (x)
+ 2g̃ V (x)χ(x) = λχ(x)
It is non-linear, since V itself depends on χ. However, it can be solved by iteration. We start with
a guess for the ground state wave function χ0 (x) and calculate V0 (x) by solving Poisson’s equation.
Using this approximate effective potential V0 (x), we solve the eigenvalue problem
−1 ′′
2µ χ1 (x)
+ g̃2 V0 (x)χ1 (x) = λχ1 (x)
for the first iterate wave function χ1 (x). χ1 (x) is used to determine a new effective potential
R∞
V1 (x) = g̃ −∞ |χ1 (x)|2 |x−y|
2 dx. (This is the solution of Poisson’s equation satisfying the boundary
conditions given in Section 3.7). The process is iterated till it converges (or until successive iterates
are the same up to numerical errors, if one is working numerically). The resulting ground state
27
χ(x) may be transformed back to momentum space and re-expressed in terms of p = k + µ. We
thus obtain an approximate ground state wave function and energy for the N parton system in the
non-relativistic limit. We shall not discuss the details here.
We mention the procedure primarily because it gives us a general method of obtaining approximate solutions to certain non-linear equations. It is essentially this procedure of iteration
that will be used in the numerical solution to our non-linear integral equation for the ground state
parton wave function. It will ofcourse be more complicated since we have a non-linear integral
equation with singularities.
4.3
Finite Part Integrals
We have mentioned the need to define singular integrals where the integrand has a
1
q2
singularity.
For instance, consider the integral equation for the ground state wave function ψ̃(p):
R
µ2
1
2 FP P Ṽ (p − q)ψ̃(q) dq = 0,
ψ̃(p)
+
g̃
(p
+
)
−
λ
2
p
2π
0
where Ṽ (p) = − p12
RP
0
dq
ψ̃ ∗ (p + q)ψ̃(q) 2π.
Hence the integrand in the equation for ψ̃(p) is singular. We will define it as a ‘Finite Part’
integral in the sense of Hadamard [19]. In general, all our singular integarls can be reduced to those
of the form
FP
RP
0
f (q)
dq
q2
where f itself is not singular at the origin. In fact, we will assume that f is C 1 at the origin from
the right. This means that f possesses a continuous right-hand derivative at the origin. We shall
rewrite the above integral as
FP
RP
0
f (q)
dq
q2
=
RP
0
f (q)−f (0)−qf ′ (0)
dq
q2
+ f (0)FP
RP
0
dq
q2
+ f ′ (0)FP
RP
0
dq
q
The first integral on the right is non-singular at the origin and is an ordinary Riemann
Integral, since we have subtracted out the singular parts. However, the last two integrals on the
RP ν
right still need to be defined. They are of the form FP 0 qq2 dq. We know that for ν > 1, this is
just a Riemann Integral and has the value
P ν−1
ν−1
We shall now define it even for ν < 1 by analytically
continuing this formula.
28
i.e.
FP
RP
0
qν
q 2 dq
=
P ν−1
ν−1
for
ν 6= 1.
This clearly runs into trouble for ν = 1. In this special case, we shall define it to be the
average of the values for ν = 1 + ǫ and ν = 1 − ǫ as ǫ → 0.
RP
This is easily seen to imply that FP 0 dq
q = Log(p). Therefore we have the following
definition:
FP
Z
FP
Z
P
qν
P ν−1
dq
=
q2
ν−1
ν 6= 1.
dq = Log(P )
ν = 1.
0
P ν
q
q2
0
We must mention a few cautionary remarks. The definition for ν = 1 violates the change of
variable formula for the scale transformation q → λq by the amount Logλ. We must take this into
account when making a scale transformation in the case ν = 1. For ν ≤ 1, the Finite Part integral
of a positive function may be negative. For example
FP
R1
dq
0 q2
= −1. and FP
R∞
dq
−∞ q 2
= 0.
However, we also note that these definitions reduce to the familiar Riemann integrals whenever they exist in the usual sense. Furthermore, the usual Cauchy Principal value for integrals with
a simple pole that does not lie at a limit of integration continues to hold. In particular then,
FP
R∞
dq
−∞ q
=0
We emphasize that these definitions are necessary since we would like to rewrite position space
potential energy integrals in terms of momentum variables. These complications arise because the
integral kernel of the Green’s function
|x|
2
expressed in momentum space takes the form
−1
.
p2
These
definitions ensure that the same integrals evaluated in position and momentum space give the
same results. For instance, the position space mean potential V (x) satisfies Poisson’s equation
V ′′ (x) = |ψ(x)|2 along with a pair of boundary conditions. Poisson’s equation in momentum space
for the mean potential Ṽ (p) is
−p2 Ṽ (p) =
RP
0
dr
ψ̃ ∗ (p + r)ψ̃(r) 2π
29
The boundary conditions translate to rules for integrating the
1
p2
singularity appearing in Ṽ (p).
As an illustration and check, let us show that our definitions do indeed imply the equality of
the Green’s functions expressed in position and momentum space. We would like to prove that
− |x|
2 = FP
R∞
1 ipx dp
2π
−∞ p2 e
The proof will involve using our definition for Finite Part Integrals since the integrand on
the right is singular. We will use contour integration to complete the proof.
For x = 0, both sides are zero according to our definition. It suffices to prove the stated
result for x > 0 since both sides are even functions of x as is easily checked.
FP
R∞
1 ipx dp
2π
−∞ p2 e
=
R∞
−∞
eipx −1−ipx dp
2π
p2
+ FP
R∞
dp
−∞ 2πp2
+ ixFP
R∞
dp
−∞ 2πp
= I1 + I2 + I3.
I3 is just the Cauchy principal value and is zero as observed before. I2 is also zero by the
Finite Part prescription. I1 is the Riemann integral of an entire function of p along the real axis.
It may be evaluated using a semicircular contour in the upper-half plane. After making some
estimates, we have I1 =
4.4
−x
2
for x > 0. Therefore, the desired result follows.
The Small p Behavior of the Valence Quark Wave Function
We shall describe a Frobenius-type analysis that gives us the behavior of the wave function ψ̃(p)
for small positive p. The integral equation for the wave function is:
RP
dq
µ2
1
2
2 (p + p ) − λ ψ̃(p) + g̃ FP 0 Ṽ (p − q)ψ̃(q) 2π = 0
RP
dq
where Ṽ (p) = − p12 0 ψ̃ ∗ (p + q)ψ̃(q) 2π.
Keeping only the singular terms for small p, we have:
1
2
2 (m
Ṽ (q) ∼
−1
q2
−
g̃ 2 1
π ) p ψ̃(p)
+ g̃FP
Rp
p−P
dq
Ṽ (q)ψ̃(p − q) 2π
=0
for small q. Now we assume a power law behavior ψ̃(p) ∼ pν for small p > 0 and
derive an equation for ν.
1 m2
2 ( g̃ 2
− π1 ) = FP
30
R1
1− Pp
(1−y)ν dy
y 2 2π
where y = pq . Since p << P , we have
πm2
g̃ 2
− 1 = FP
R1
0
(1−y)ν +(1+y)ν
dy
y2
+
R∞
1
(1+y)ν
dy
y2
The first of these integrals is singular and we evaluate it according to the definition given
earlier. The result is a transcendental equation for ν:
πm2
g̃ 2
−1=
R1
0
(1+y)ν +(1−y)ν −2
dy
y2
−2+
R∞
1
(1+y)ν
y 2 dy
πm2 /g̃2 − 1 = h(ν). It is easily seen that for m = 0, ν = 0 is a solution. In
which is of the form
the limit of zero current quark mass, the critical wave function tends to a constant at the origin.
Calculating h(ν) shows that for a positive quark mass m, the wave function goes to zero at p = 0
and rises like a power law ψ̃(p) ∼ pν , ν > 0. The following plot shows h(ν). The solution ν for a
2
− 1 intersects the curve.
given value of m ≥ 0 is the point at which the horizontal line π m
g̃ 2
0.1
0.2
0.3
0.4
0.5
-0.2
-0.4
-0.6
-0.8
-1
Figure 5. h(ν) as a function of ν. The solution ν for a given value of m ≥ 0 is the point at which
2
− 1 intersects the curve. Notice that for m = 0, ν = 0 is a solution: in the
the horizontal line π m
g̃ 2
limit of Chiral Symmetry, the wave function goes to a non-zero constant at the origin.
Our conclusion that the critical momentum space wave function tends to a non-zero constant
as p → 0 implies that it is discontinuous at the origin. Therefore, the critical position space wave
function decays like
1
x
at infinity. This is very reasonable from another point of view. In the soliton
model [12] the baryon is thought of as being made up of an infinite number of pions. It has been
shown that this picture is consistent with QCD [13]. But in that case, it should be possible to
31
approximate the baryon wave function with the meson wave function for large spatial x. At large
distances we are essentially far away from the proton. The meson wave function has been calculated
by ’t Hooft [10]. In the limit m → 0 the ’t Hooft meson wave function too has a discontinuity in
momentum space and consequently decays like
1
x
as x → ∞, just as we obtain here.
This Frobenius-like analysis is very useful. It gives us a clear quantitative understanding of
the small p behavior of the wave function in the physically interesting region m → 0. We have
that as m → 0, ψ̃(p) ∼ pν with ν → 0. i.e. the wave function rises sharply near the origin. At the
critical point m = 0, the wave function goes to a non-zero constant at p = 0. We had already got a
scent of this behavior from our variational estimates! Together with the numerical solution, which
will be valid except for small p, we will have almost a complete understanding of the valence quark
wave function.
4.5
Numerical Solution
The numerical solution of this problem is not straightforward since the kernel of the integral equation
is singular. We need a reliable method of numerical quadrature for integrals such as
FP
RP
0
f (p, q)ρ(q)dq
when the weight function ρ(q) has a singularity at q = 0 like
the interval
points
1
q2 .
We need to subdivide
[0, P ] = ∪nr=1 [br , br+1 ] into subintervals. Within each subinterval we choose a set of
qjr , j = 1, · · · νr . We approximate the integral by a sum
FP
RP
0
f (p, q)ρ(q)dq =
P
jr
wjr f (qjr ).
The weights wjr are determined by the condition that within each subinterval [br , br+1 ] ,
the integral of a polynomial of order
FP
νr − 1 is reproduced exactly:
R br+1
br
q k ρ(q)dq =
Pνr
k
j=1 wjr qjr.
This is the usual method of numerical quadrature [19, 20] except that the integral on the
left hand side is singular for
br = 0 and
k = 0, 1. In these cases we can evaluate the left hand
side analytically as the finite part in the sense of Hadamard. (The main difference from the usual
32
situation is that the moments on the left hand side of the above equation are not all positive.) The
weights are then determined by solving the above system of linear equations.
Given an approximate mean potential
1
2 [p
+
µ2
p
V˜s (p) we can convert the linear integral equation
− 2λs ]ψ̃s+1 (p) + g̃2 FP
R∞
0
dq
V˜s (p − q)ψ̃s+1 (q) 2π
=0
into a matrix eigenvalue problem by the above method of quadrature. We use the ground
state eigenfunction so determined to calculate numerically the next approximation
Ṽs+1 for
the mean potential. This process is iterated until the solution converges. Having determined the
wavefunction, we must impose the momentum sum rule to determine
g̃. We used Mathematica
to implement this numerical procedure. An approximate analytic solution was used as a starting
point for the iteration.
4.6
Comparison with Experimental Data
Now we turn to the question of the comparison of our model with data from Deep Inelastic Scattering. It is customary to describe the parton distributions as functions of the Bjorken scaling variable
0 ≤ xB ≤ 1 which is the fraction of the null component of momentum carried by each parton. This
means we must rescale momenta to the dimensionless variable xB =
p
P.
The probability density of
a parton carrying a fraction xB of the momentum is then
φ(xB ) =
(The factor of
rather than
1
2π
P
2
2π |ψ̃(xB P )| .
is needed because φ(xB ) is traditionally normalized to one with the measure dxB
dxB
2π .)
It is important to note that the only dimensional parameter in our theory, g̃, cancels out of
the formula for φ(xB ): it only serves to set the scale of momentum and when the wavefunction
is expressed in terms of the dimensionless variable xB , it cancels out. The wave function only
depends on the ratio
m2
g̃ 2
and is independent of g̃ in the limit of zero current quark mass. We have
set µ2 to the critical value (within numerical errors), which is the value corresponding to chiral
symmetry; i.e., zero current quark mass. The number of colors we fix at N = 3. Thus the only free
parameter in our theory is the fraction f of the baryon momentum carried by the valence partons.
The parameters N and f appear in the combination Neff =
33
N
f .
We have ignored the ispospin of the quarks in the above discussion. We should therefore
compare our structure functions with the isoscalar combination of the valence quark distributions
of a baryon, φ(xB ). It is not difficult to take into account the effects of isospin.
The parton distributions have been extracted from scattering data by several groups of
physicists [4, 5, 6]. In the figure below we plot our wavefunctions and compare them to that
extracted from data by the MRST collaboration, at Q2 = 1 GeV2 . We agree remarkably well
with data except for small values of xB . The agreement is best when the fraction of the baryon
momentum carried by the valence partons is about f ∼ 0.5.
Wave Functions Versus xB
3
<---MRST fit to DATA
2.5
2
<---Numerical f~.6
1.5
<---Analytic f~.5
1
0.5
0.2
0.4
0.6
0.8
Figure 1. Comparison of the predicted valence parton wavefunction
1
√
xB
φ(xB ) with the MRST [5]
global fit to data. The wavefunction we predict goes to a non-zero constant at the origin. The
numerical solution is not reliable in this region. The ‘analytic’ wave function is the variational
estimate that minimizes the energy. The fit to data has a mild divergence at the origin.
Our model does not predict the observed behavior of the parton distributions for small xB :
our probability distribution tends to a constant for small xB (Sections 4.1, 4.4) although due to
numerical errors this is not evident in the numerical solution. The observed distributions have an
for small xB .The approximations
integrable singularity there: roughly speaking, φ(xB ) ∼ x−0.5
B
we made clearly break down in the small xB region: travsverse momenta and sea quarks can no
longer be ignored, indeed even gluons need to be considered. We will study these effects in future
publications.
34
5
Conclusion
This thesis has been about the structure of the proton. The Deep Inelastic Scattering experiments
showed that the proton is made of point-like constituents, called quarks or partons. The structure
of the proton is described in terms of parton distribution functions. These are momentum-space
probability distributions of quarks inside the proton. Parton distribution functions are the analogues of electron wave functions in an atom, and are of great importance in particle physics. They
are measured in Deep Inelastic Scattering experiments. However, we do not yet have a theoretical understanding of the xB (momentum fraction or Bjorken Scaling variable) dependence of the
parton distribution functions.
In this thesis we solve the problem of predicting valence quark distributions in the proton.
We have presented a model of interacting partons for the structure of baryons. The valence quarks
interact through a linear potential in the null coordinate. This model can be derived from QCD
in the approximation where transverse momenta are ignored. We obtain the valence quark wave
function as the solution to a non-linear integral equation. We understand the behavior of the
solution both analytically and numerically. Our prediction is compared with the baryon structure
function extracted from global fits to Deep Inelastic Scattering data. The only parameter we can
adjust is the fraction of baryon momentum carried by the valence partons. Our predictions agree
well with data except for small values of the Bjorken Scaling variable, xB , where our model is not
expected to be accurate. We cannot ignore gluons, sea quarks and transverse momenta. We shall
study these effects in future publications.
35
Acknowledgements
The author would like to thank his advisor Professor S. G. Rajeev for the wonderful opportunity
to work on this problem and his insightful guidance. The author also thanks research associate
Varghese John for useful discussions.
This research was supported by the Department of Energy through the grant DE-FG0291ER40685 and the Summer Reach program of the University of Rochester.
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